Talk:Polar set

Polar set is not defined
Currently, the article does not say what a polar set is. It just says that a polar set is a set in the dual space if it is a polar set of a subset of a vector space. -- Kjkolb 01:09, 7 May 2007 (UTC)


 * I tried to clarify the article. MathMartin 16:57, 8 May 2007 (UTC)

Theorem of bipolars
I suggest adding something like the following

``* By the theorem of bipolars $$A^{00}$$ is convex weak-closure of $$A \cup \{0\}$$, that is the smallest weak-closed convex set containing $$A$$ and 0.''

This asks for right links to the definition of weak topology, sometimes denoted as $$\sigma(X,Y)$$ Matumba (talk) 01:27, 27 April 2008 (UTC)

As it stands, the article statement is wrong: $$A^{00}$$ is not equal to the absolutely convex envelope (take the open unit ball in the reals for example), but to the weak-closure of it. Chrystomath2 (talk) 09:03, 15 April 2016 (UTC)

Confusion

 * The page says "There are at least three competing definitions of the polar of a set.", lists two of them, and then talk about properties. Does these properties hold for all definitions or what is intended?


 * The page says "Some authors (confusingly) call a dual cone the polar cone; we will not follow that convention in this article.", and then makes this definition in the last property: "For a closed convex cone C, the dual cone is the polar of C". — Preceding unsigned comment added by 5.57.55.92 (talk) 10:34, 24 January 2020 (UTC)


 * And I would add, it is not very smart, and quite confusing, adopting the same name and notation for the polar of the polar (i.e. the bi-polar) of $$A\subset X$$ and the pre-polar of the polar. Polar sends to the dual, pre-polar sends backwards to the pre-dual. The polar of a subset of $$B\subset X^*$$ is a subset of $$X^{**}$$; the pre-polar is just the trace of its polar on $$X$$, via the isometric embedding $$X\to X^{**}$$. Also, the weak topology of $$X$$ ($$\sigma(X,X^*)$$) is just the trace on $$X$$ of the weak* topology of  $$X^{**}$$ ($$\sigma(X^{**},X^*)$$).  pm a  23:41, 9 November 2022 (UTC)